AP®︎ Calculus AB (2017 edition)
Course: AP®︎ Calculus AB (2017 edition) > Unit 2Lesson 4: Formal definition of the derivative as a limit
- Formal definition of the derivative as a limit
- Formal and alternate form of the derivative
- Worked example: Derivative as a limit
- Worked example: Derivative from limit expression
- Derivative as a limit
- The derivative of x² at x=3 using the formal definition
- The derivative of x² at any point using the formal definition
The derivative of x² at any point using the formal definition
Sal finds the limit expression for the derivative of f(x)=x² at any point x and simplifies it into a simpler expression (spoiler: it's 2x). Created by Sal Khan.
Want to join the conversation?
- Why does he call the lines Tangent and Secant? What is the relation to Trig?(34 votes)
- Tangent is the line that 'touchs' the curve
Secant is the line that 'cuts' the curve
The curve is also sometimes called 'sine' (From latin 'sinus' meaning curve)
Full explanation of relation to trig: http://mathforum.org/library/drmath/view/54053.html(51 votes)
- could you ever have a Derivative being "no slope" I know that a slope of zero is the X axis, but isn't a line of no slope the Y axis? Is my question of merit? or is it nonsesne?(27 votes)
- When a line or curve has a slope parallel to the y-axis, the slope is undefined, because you are dividing by zero. If you have a curve with a vertical tangent line at one point, you will see the derivative approach infinity for that point.(13 votes)
- 'd' doesn't stand for delta, does it?
That would be too simple(18 votes)
- The 'd' in fact does come from delta(Δ). dy/dx is similar to Δy/Δx which is the slope formula (that is rise over run).
Δy/Δx =(y2-y1)/(x2-x1) but in dy/dx the difference between the two points like x2 and x1 is taken to be much smaller or more accurately using limits to approach 0 for getting the slope at a single point.
The derivative so gives the slope of a tangent line that touches a curve only once. The slope formula in contrast gives the slope of a secant ( a line that intersects a curve twice) through a curve as it uses two points with significant difference.
The small delta symbol (δ) is also sometimes used to show a very very small change.(23 votes)
- At7:20, why does h disappear because one takes the limit as h approaches 0?(10 votes)
- That is a result of the Direct Substitution Property which says "If f is a polynomial or a rational function and a is in the domain of f, then lim x->a f(x) = f(a)"
In this case, x is replaced by h and a is zero, so h disappears because we substitute 0 for h. That is why it was necessary to "clear" h from the denominator or the difference quotient. So that when we do direct substitution, we do not end up dividing by zero, which is undefined.(10 votes)
- Where does the word 'calculus' come from?(5 votes)
- It's true; the word calculus means pebble in Latin. In ancient Rome, pebbles were used to perform arithmetic (they didn't have calculus or even algebra), so this also gives us the word "calculate." Doing any math become closely associated with pebbles, and as a result, we got the word calculate and we began calling a specific type of math calculus even though calculus as we know it today never had anything to do with pebbles.(14 votes)
- I am wondering if derivative is a slope of some function, couldn't we put that slope in a general function formula to form a function which derivative we took? Something like this f(x)=f '(x) * x + b
But then 2x*x should equal x^2, but it does not. Where am I wrong?(6 votes)
- And now this is very useful. I just realized these two topics aren't in the videos so I suggested it.
1. THis is useful for something we called linear approximation. We know the tangent lines so we can now approximate some close x values. This is very useful when you don't have a calculator and also for newton's method which is point 2.
f(x) ~ f(a) + f'(a) (x-a)
a would be a nice close number.
2. It's used for newton's method.
This is a method used for finding zeroes. Sometimes zeroes are very hard to find without a graphing calculator. You just can't use algebra to find it. SO this approximation is a algorithm thing. SO you keep doing again and again and usually you get closer to the exact number.(5 votes)
- at6:30, how does (x+h)^2 become (x^2) + 2xh + (h^2) ?(4 votes)
- (x+h)^2 = (x+h) * (x+h) Multiplying these together you get (x^2)+2xh +(h^2)(7 votes)
- At07:00, we simplify
lim h->0 [ (2xh + h^2) / h ]
by dividing the numerator by the denominator, h, to leave
lim h->0 [ 2x + h ].
In some of the previous limit videos, when we had a variable in the denominator that approached zero, we had to specify that the limit would be true so long as the variable was not ACTUALLY equal to zero. Why do we not specify that h is not equal to zero in this case?(5 votes)
We do not actually SET h = 0, we just make h TEND to 0 so divisibility by h is acceptable.(3 votes)
- So is there just a universal formula you can use to solve for any of the points or do you have to plug and chug based on what you get from any particular graph?(3 votes)
- f(x +h) - f(x)/ h as h approaches zero is the universal formula for finding the slope of any given function. It is not just used for Power functions. It then can be simplified to find the Rules of Derivative. This formula is the base of these rules.(3 votes)
- Is there any difference between dy/dx and d/dx?(3 votes)
- dy/dx yields a function, namely the derivative of y(x) (y as a function of x). d/dx is more of an operator. If you "multiply" d/dx by a function, you will get the derivative of that function. For example d/dx(y)=dy/dx. The notation may look different, but it is the same thing.(3 votes)
In the last video, we found the slope at a particular point of the curve y is equal to x squared. But let's see if we can generalize this and come up with a formula that finds us the slope at any point of the curve y is equal to x squared. So let me redraw my function here. It never hurts to have a nice drawing. So that is my y-axis. That is my x-axis right there. My x-axis. Let me draw my curve. It looks something like that. You've seen that multiple times. This is y is equal to x squared. So let's be very general right now. Remember, if we want to find-- let me just write the definition of our derivative. So if we have some point right here-- let's call that x. So we want to be very general. We want to find the slope at the point x. We want to find a function where you give me an x and I'll tell you the slope at that point. We're going to call that f prime of x. That's going to be the derivative of f of x. But all it does is, look, f of x, you give-- it's a function that you give it an x, and it tells you the value of that. And we draw the curve here. With f of x, you give that same x but it's not going to tell you the value of the curve. It's not going to say, oh, this is your f of x. It's going to give you the value of the slope of the curve at that point. So f of x, if you put it into that function, it should tell you, oh, the slope at that point, is equal to-- you know, if you put 3 there, you'll say, oh, the slope there is equal to 6. We saw that in the last example. So that's what we want to do. And we saw on the last-- I think it was 2-- videos ago, that we defined f prime of x to be equal to-- just the-- well, I'll write it this way. It's the slope of the secant line between x and some point that's a little bit further away from x. So the slope of the secant line is change in y. So it's the y value of the point that's a little bit further away from x. So f of x plus h minus the y value at x, right? Because this is right here. This is f of x. So minus f of x. All of that over the change in x. So if this is x plus h here, the change in x is x plus h minus x. Or this distance right here is just h. The change in x is going to be equal to h. So that's just slope of the secant line, between any 2 points like that. And we said, hey, we could find the slope of the tangent line if we just take the limit of this as it approaches-- as h approaches 0. And then we'll be finding the slope of the tangent line. Now let's apply this idea to a particular function, f of x is equal to x squared. Or y is equal to x squared. So here, we could have the point-- we could consider this to be the point x-- x squared. So f of x is just equal to x squared. And then this would be the point-- let me do it in a more vibrant color. This is the point x plus h-- that's this point right here. It's a little bit further down. And then x plus h squared. And you know, in the last video, we did this for a particular x. We did it for 3. But now I want a general formula. You give me any x and I won't have to do what I did in the last video for any particular number. I'll have a general function. You give me 7, I'll tell you what the slope is at 7. You give me negative 3, I'll tell you what the slope is at negative 3. You give me 100,000, I'll tell you what the slope is at 100,000. So let's apply it here. So we want to find the change in y over the change in x. So first of all, the change in y is this guy's y value, which is x plus h squared. That's this guy's y value right here. That's this right here. That's x plus h squared. I just took x plus h, evaluated, I squared it, and that's its point on the curve. So it's x plus h squared. So that's there right there. And then what's this value? f of x right here is equal to-- I know it's getting messy-- is equal to x squared. If you take your x, you evaluate the function at that point, you're going to get x squared. So it's equal to minus x squared. This is your change in y. That's this distance right there. And just to relate it to our definition of a derivative, this blue thing right here is equivalent to this thing right here. We just evaluated our function. Our function is f of x is equal to x squared. We just evaluated when x is equal to x plus h. So if you have to square it, if I put an a there, it'd be a squared. If I put an apple there, it'd be apple squared. If I put an x plus h in there, it's going to be x plus h squared. So this is that thing. And then, this thing right here is just the function evaluated at the point in question. Right there. So this is our change in y. And let's divide that by our change in x. Our change in x-- if this is x plus h and this is just x, our change in x is just going to be h. So that's where we get that term from. So this is just a slope between these 2 points. This is just a slope between those two points. But, of course, we want to find-- the limit at this point gets closer and closer to this point, and this point gets closer and closer to that point. So this becomes a tangent line. So we're going to take the limit as h approaches 0, and this is our f prime of x. And this is the exact same definition of this, instead of being general and saying, for any function, we know what the function was. It was f of x is equal to x squared. So we actually applied it. Instead of f of x, we wrote x squared. Instead of f of x plus h, we wrote x plus h squared. So let's see if we can evaluate this limit. So this is going to be equal to the limit as h approaches 0 to square this out. I'll do it in the same color. That's x squared plus 2xh plus h squared, and then we have this minus x squared over here. I just multiplied this guy out over here. And then all of that is divided by h. Now let's see if we can simplify this a little bit. Well, you immediately see you have an x squared and you have a minus x squared, so those cancel out. And then we can divide the numerator and the denominator by h. So this simplifies to-- so we get f prime of x is equal to-- if we divide the numerator and the denominator by h-- we get 2x plus h. I'm sorry, I forgot my limit. It equals the limit. Very important. Limit as h approaches 0 of divide everything by h, and you get 2x plus h squared divided by h is h. And if you remember the last video, when we did it with a particular x, we said x is equal to 3, we got 6 plus delta x here. Or 6 plus h here, so it's very similar. So if you take the limited h approaches 0 here, that's just going to disappear. So this is just going to be equal to 2x. So we just figured out that if f of x-- this is a big result. This is exciting! That if f of x is equal to x squared, f prime of x is equal to 2x. That's what we just figured out. And I wanted to make sure you understand how to interpret this. f of x, if you give me a value, is going to tell you the value of the function at that point. At prime of x it's going to tell you the slope at that point. Let me draw that. Because this is a key realization. And you might, you know, it's kind of maybe initially unintuitive to think of a function that gives us the slope, at any point, of another function. So it looks like this. Let me draw a little neater than that. Ah, it's still not that neat. That's satisfactory. Let me just draw it in the positive coordinate. Well, I'll just draw the whole-- the curve looks something like that. Now this is the curve of f of x. This is the curve of f of x is equal to x squared. Just like that. So if you give me a point. You give me the point 7. You apply, you put it in here, you square it. And it is mapped to the number 49. So you get the number 49 right there. This is the number 7, 49. You're used to dealing with functions right there. But what is f prime of 7? f prime of 7. You say, 2 times 7 is equal to 14. What is this 14 number here? What is this thing? Well, this is the slope of the tangent line at x is equal to 7. So if I were to take that point and draw a tangent line-- a point that just grazes our curve-- if I were to just draw a tangent line. That wasn't tangent enough for me. So that's my tangent line right there. You get the idea. The slope of this guy-- you do your change in y over your change in x-- is going to be equal to 14. The slope of the curve at y is equal to 7-- is a pretty steep curve. If you wanted to find the slope, let's say that this is y-- let's say it's x is equal to 2. I said at x is equal to 7, the slope is 14. At x is equal to 2, what is the slope? Well, you figure out f prime of 2, which is equal to 2 times 2, which is equal to 4. So the slope here is 4. You could say m is equal to 4. m for slope. What is f prime of 0? f prime. We know that f of 0 is 0, right? 0 squared is 0. But what is f prime of 0? Well, 2 times 0 is 0. That's also equal to 0. But what does that mean? What's the interpretation? It means the slope of the tangent line is 0. So a 0 sloped line looks like this. Looks just like a horizontal line. And that looks about right. A horizontal line would be tangent to the curve at y equals 0. Let's try another one. Let's try the point minus 1. So let's say we're right there. x is equal to minus 1. So f of minus 1, you just square it. Because we're dealing with x squared. So it's equal to 1. That's that point right there. What is f prime of minus 1? f prime of minus 1 is 2 times minus 1. 2 times minus is minus 2. What does that mean? It means that the slope of the tangent line at x is equal to 1, to this curve, to the function, is minus 2. So if I were to draw the tangent line here-- the tangent line looks like that-- and look, it is a downward sloping line. And it makes sense. The slope here is equal to minus 2.